arXiv subjects
Publications and source records attributed to Azahara DelaTorre.
Given $s\in(0,1)$ and $n>2s$, we construct nontrivial solutions of the problem \begin{equation*} \begin{cases} (-Δ)^s u=u^{2^*_s-1} &{\mbox{ in }}\mathbb{R}^n_+,\\ {\mathcal{N}}_s u=0&{\mbox{ in }}\mathbb{R}^n_-,\end{cases} \end{equation*}where ${\mathcal{N}}_s$ represents the nonlocal Neumann condition of exterior type.
We derive Kazdan-Warner type identities for the boundary problem of prescribing nonconstant interior $Q$ curvature and boundary $T$ curvature on the upper hemisphere $\mathbb{S}^4_+$ by a conformal change of the standard metric. Using the natural variational formulation and conformal variations generated by boundary-preserving conformal vector fields, we obtain nontrivial integral obstructions to solvability.
We prove the non-degeneracy of solutions to a fractional and singular Liouville equation defined on the whole real line in presence of a singular term. We use conformal transformations to rewrite the linearized equation as a Steklov eigenvalue problem posed in a bounded domain, which is defined either by an intersection or a union of two disks. We conclude by proving the simplicity of the corresponding eigenvalue.
In this paper we investigate the validity of Hopf's Lemma for a (possibly sign-changing) function $u \in H^s_0(Ω)$ satisfying \[ (-Δ)^s u(x) \geq c(x)u(x) \quad \text{in }Ω,\] where $Ω\subset \mathbb{R}^N$ is an open, bounded domain, $c \in L^\infty(Ω)$, and $(-Δ)^s u$ is the fractional Laplacian of $u$. We show that, under suitable assumptions, the validity of Hopf's Lemma for $u$ at a point $x_0 \in \partial Ω$ is essentially equivalent to the validity of Hopf's Lemma for the Caffarelli-Silvestre extension of $u$ at the point $(x_0,0) \in \mathbb{R}^N \times \mathbb{R}^+$. We also provide a slightly more precise characterization of a dichotomy result stated in a recent paper by Dipierro, Soave and Valdinoci.
The Delaunay metrics form a family of conformally flat, constant fractional Q-curvature metrics on a twice-punctured sphere. They are all (after a Möbius transformation) rotationally symmetric and periodic, and admit several elegant variational descriptions. We prove that, when s is close to but less than 1, any complete, conformally flat constant Q-curvature metric on a twice-punctured sphere is a Delaunay metric. Along the way, we prove a sharp a priori bound for the conformal factor of these metrics, which may be of independent interest.
We consider the first positive Steklov eigenvalue on planar domains. First, we provide an example of a planar domain for which a first eigenfunction has a closed nodal line. Second, we establish a lower bound for the first positive eigenvalue on certain symmetric domains and show that this eigenvalue is simple for all ellipses. These results complement two statements contained in a work by Kuttler and Sigillito (Proc. Amer. Math. Soc. 20, 1969).
In this paper, we address the problem of prescribing non-constant $Q$ and boundary $T$ curvatures on the upper hemisphere $\mathbb{S}^4_+\subset \mathbb{R}^5$, via a conformal change of the background metric. This is equivalent to solve a fourth-order non-linear elliptic boundary value problem with a third-order non-linear equation and homogeneous Neumann conditions at the boundary. The problem admits a Mean-field type variational formulation, similar to the one obtained by Cruz-Blázquez and Ruiz for a related problem in two dimensions, with the associated energy functional being bounded from below but, in general, not coercive. By imposing symmetry conditions, we are able to prove the existence of minimizers, especially when $Q,T\geq 0$. To the best of our knowledge, these are the first existence results obtained for this setting.
Point singularities of solutions to the classical Lane-Emden-Serrin equation have a polyhomogeneous asymptotic expansion whose logarithmic corrections are determined by a first order ODE. Surprisingly, we are able to discover such an ODE for the fractional Lane-Emden-Serrin equation, and therefore give a short classification for the precise local behavior of its solutions up to the second order involving a double logarithm. This seems to be the first time that a nonlocal equation is associated to a genuinely local ODE in one dimension. New non-existence, existence and multiplicity results for the corresponding Dirichlet problem are also discussed. Moreover, we construct complete $s$-fractional Yamabe metrics in $\mathbb{R}^n$ which are singular along a smooth submanifold of dimension $(n-2s)/2$, via direct integral asymptotic analysis with global geometric weights. This covers the missing borderline case as suggested by the deep work of Schoen and Yau. While such dimension is maximal in the class of distributional solutions, we conjecture the existence of complete metrics, understood in a suitably generalized sense, with prescribed singularities of strictly higher dimensions.
We prove uniqueness of least-energy solutions to the fractional Lane-Emden equation, under homogeneous Dirichlet exterior conditions, when the underlying domain is a ball $B \subset \mathbb{R}^N$. The equation is characterized by a superlinear, subcritical power-like nonlinearity. The proof makes use of Morse theory and is inspired by some results obtained by C. S. Lin in the '90s. A new Hopf's Lemma-type result shown in this paper is an essential element in the proof of nondegeneracy of least-energy solutions.
In this paper we are interested in the qualitative properties of the solutions to the fractional Yamabe problem in $\mathbb{R}^n$ which present an isolated singularity. In particular, we prove that the Morse index of any such solution is infinity. The proof uses a Emden Fowler type transformation, so that we can pass to a nonlocal 1D problem posed in $\mathbb{R}$.
In this paper, we will consider the fractional Caffarelli-Kohn-Nirenberg inequality \begin{equation*} Λ \left(\int_{\mathbb R^n}\frac{|u(x)|^{p}}{|x|^{β {p}}}\,dx\right)^{\frac{2}{p}}\leq \int_{\mathbb R^n}\int_{\mathbb R^n}\frac{(u(x)-u(y))^2}{|x-y|^{n+2γ}|x|^{α}|y|^{α}}\,dy\,dx \end{equation*} where $γ\in(0,1)$, $n\geq 2$, and $α,β\in\mathbb R$ satisfy \begin{equation*} α\leq β\leq α+γ, \ -2γ<α<\frac{n-2γ}{2}, \end{equation*} and the exponent $p$ is chosen to be \begin{equation*} p=\frac{2n}{n-2γ+2(β-α)}, \end{equation*} such that the inequality is invariant under scaling. We first study the existence and nonexistence of extremal solutions. Our next goal is to show some results on the symmetry and symmetry breaking region for the minimizers; these suggest the existence of a Felli-Schneider type curve separating both regions but, surprisingly, we find a novel behavior as $α\to -2γ$. The main idea in the proofs, as in the classical case, is to reformulate the fractional Caffarelli-Kohn-Nirenberg inequality in cylindrical variables. Then, in order to find the radially symmetric solutions we need to solve a non-local ODE. For this equation we also get uniqueness of minimizers in the radial symmetry class; indeed, we show that the unique continuation argument of Frank-Lenzmann (Acta'13) can be applied to more general operators with good spectral properties. We provide, in addition, a completely new proof of non-degeneracy which works for all critical points. It is based on the variation of constants approach and the non-local Wronskian of Ao-Chan-DelaTorre-Fontelos-González-Wei (Duke'19).
We study the existence of sign-changing solutions for a non-local version of the sinh-Poisson equation on a bounded one-dimensional interval $I$, under Dirichlet conditions in the exterior of $I$. This model is strictly related to the mathematical description of galvanic corrosion phenomena for simple electrochemical systems. By means of the finite-dimensional Lyapunov-Schmidt reduction method, we construct bubbling families of solutions developing an arbitrarily prescribed number sign-alternating peaks. With a careful analysis of the limit profile of the solutions, we also show that the number of nodal regions coincides with the number of blow-up points.
We answer affirmatively a question of Aviles posed in 1983, concerning the construction of singular solutions of semilinear equations without using phase-plane analysis. Fully exploiting the semilinearity and the stability of the linearized operator in any dimension, our techniques involve a careful gluing in weighted $L^\infty$ spaces that handles multiple occurrences of criticality, without the need of derivative estimates. The above solution constitutes an \emph{Ansatz} for the Yamabe problem with a prescribed singular set of maximal dimension $(n-2)/2$, for which, using the same machinery, we provide an alternative construction to the one given by Pacard. His linear theory uses $L^p$-theory on manifolds, while our approach studies the equations in the ambient space and is therefore suitable for generalization to nonlocal problems. In a forthcoming paper, we will prove analogous results in the fractional setting.
Non-local equations cannot be treated using classical ODE theorems. Nevertheless, several new methods have been introduced in the non-local gluing scheme of our previous article "On higher dimensional singularities for the fractional Yamabe problem: a non-local Mazzeo-Pacard program"; we survey and improve those, and present new applications as well. First, from the explicit symbol of the conformal fractional Laplacian, a variation of constants formula is obtained for fractional Hardy operators. We thus develop, in addition to a suitable extension in the spirit of Caffarelli--Silvestre, an equivalent formulation as an infinite system of second order constant coefficient ODEs. Classical ODE quantities like the Hamiltonian and Wrońskian may then be utilized. As applications, we obtain a Frobenius theorem and establish new Poho\vzaev identities. We also give a detailed proof for the non-degeneracy of the fast-decay singular solution of the fractional Lane-Emden equation.
We consider the problem of constructing solutions to the fractional Yamabe problem that are singular at a given smooth sub-manifold, and we establish the classical gluing method of Mazzeo and Pacard for the scalar curvature in the fractional setting. This proof is based on the analysis of the model linearized operator, which amounts to the study of an ODE, and thus our main contribution here is the development of new methods coming from conformal geometry and scattering theory for the study of non-local ODEs. No traditional phase-plane analysis is available here. Instead, first, we provide a rigorous construction of radial fast-decaying solutions by a blow-up argument and a bifurcation method. Second, we use conformal geometry to rewrite this non-local ODE, giving a hint of what a non-local phase-plane analysis should be. Third, for the linear theory, we examine a fractional Schrödinger equation with a Hardy type critical potential. We construct its Green's function, deduce Fredholm properties, and analyze its asymptotics at the singular points in the spirit of Frobenius method. Surprisingly enough, a fractional linear ODE may still have a two-dimensional kernel as in the second order case.
We study the compactness properties of metrics of prescribed fractional $Q$-curvature of order $3$ in $\R^3$. We will use an approach inspired from conformal geometry, seeing a metric on a subset of $\R^3$ as the restriction of a metric on $\R^4_+$ with vanishing fourth-order $Q$-curvature. We will show that a sequence of such metrics with uniformly bounded fractional $Q$-curvature can blow up on a large set (roughly, the zero set of the trace of a nonpositive biharmonic function $Φ$ in $\R^4_+$), in analogy with a $4$-dimensional result of Adimurthi-Robert-Struwe, and construct examples of such behaviour. In doing so, we produce general Poisson-type representation formulas (also for higher dimension), which are of independent interest.
We consider the fractional mean-field equation on the interval $I=(-1,1)$ $$(-Δ)^\frac{1}{2} u=ρ\frac{e^{u}}{\int_{I}e^{u}dx},$$ subject to Dirichlet boundary conditions, and prove that existence holds if and only if $ρ<2π$. This requires the study of blowing-up sequences of solutions. We provide a series of tools in particular which can be used (and extended) to higher-order mean field equations of non-local type.
We construct solutions for the fractional Yamabe problem that are singular at a prescribed number of isolated points. This seems to be the first time that a gluing method is successfully applied to a non-local problem. The main step is an infinite-dimensional Lyapunov-Schmidt reduction method, that reduces the problem to an (infinite dimensional) Toda type system.